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eCJlH

P;JIH

Hee

BlilnOJlHHeTCH

COOTHOllIeHHe

(57.6).

11

IlYCTb

Tenep.b

MIil·

HMeeM

nOCJIep;OBaTeJlbHOCTb

.pyHKuHH

h(i)

('t)

(i=

1,

2,

...

),

3ap;aHHlilx

Ha

oTpe3Ke

[tQt,

t

13

J.

rOBOpHT,

qTO

IIOCJlep;Cl-

BaTeJl

hHOCTb

h(i)('t)

cxooumcR, K

.pyHKUHH

hO('t)

H3

,AI'

no

HopMe

p[hJ.

eCJlH

li~

P

[h(iJ

('t)

-

hO

('t)

J

==;0

O.

(57.7)

iIi

i-oo

,t

~'1

(J:>yHKUHIO

hO

(-r)

,Ha3hIBaIOT

npe8e.ltoJrt

nOCJleAOBaTeJIbHOCTH

h(i)

('t).

RaK

liI B

cJlyqae

qHCJIOBOH

nOCJlep;OBaTeJlhHOCTH,

nOCJlep;OBaTeJlbHOCTb

h(i)

(or)

Ha3hIBaeTCH

if;ynoaJrtenma.ttblf,OU,

eCJlH

BlilnOJlHHeTCH

COOTHO-

llIeHliIe

.

p

[h

li

h

Ul

I~I

lim

)

('t)

-

('t)

J =

d.

(57.8)

i,j~

,

~i

,ll,JIH

HOpMliIpoBaHHlilx

.pYHKuHoHaJlbHhlx

npOCTpaHCTB

liI3

CX~

P;liIMOCTliI

IIOCJlep;OBaTeJlhHOCTliI

hli)('t)

CJlep;yeT,

LITO

OHa

HBJlHeTCH

'i

/

'f'

.pYHp;aMeHTaJlbHOH,

.T.

e.

paBeHcTBo

(57.7)

BJleqeT

3a

C060H

BlilnOJI-

/III

HeHHe

COOTHOllIeHHH

(57.8).

Op;HaKo

He

P;JlH

BCHKoro

HOpMHpOBaHHoro

npOCTpaHCTBa

JII06aH

.pyH)J,aMeHTaJlhHaH

nOCJlep;OBaTeJlbHOCTb

HMepr

npep;eJIhO

('t)

H3

Toro

)-Ke

caMoro

npOCTpaHCTBa.

1l0aTOMy

nOJle3HO

Bbl-

p;eJlHTb

Te

.pYHKUliIOHaJlbHlile

npOCTpaHCTBa,

P;JIH

KorophIX

nplilHUHU

CXOP;HMOCTliI

ROIliH

cnpaBep;JlHB

He

TOJlbKO

B

CBoeH

He06xop;liIMOH,

HO

H B -p;OCTaTOqHOH

qaCTH.

B

CBH3H

C

aTHM

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npOCTpaHCTBO

,AI'

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nO.ltlf,b/,Jrt,

eCJlH

B

HeM

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rrOCJIe-

p;OBaTeJlbHOCTb

h(i)('t)

HMeeT

rrpep;eJIOM

.pyHKUHIO

hO

('t)

H3

Toro

me

caMoro

npOCTpaHCTBa

.;V'.

ROHeqHOMepHoe

BeKTopHoe

npOCTpaHCTBO

06H3aTeJIbHO

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rrOJIHhIM,

TaK

KaKB

rrpOCTpaHCTBe

:Jil

n

JlI06aH

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rrOCJlep;OBaTeJlhHOCTb

HMeeT

npep;eJI.

IlpocTpaH

CTBa

2(2),

2,

~,

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TaKme

Bce

p;pyrHe

rrpOCTpaHCTBa

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KO-

!

TophIe

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B

aTOH

KHHre,

·TaKme

rrOJIHhI.

Op;HaKo

p;OKa3aTeJlb-

CTBO

aTOH

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3a

paMKH

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!

aaTeJlbCTBO

::noro

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B

Kypcax

[9*,

14*

I)

.

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h ('t),

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[t""

t131,

6y~eM

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t]

B

TeX

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Qt,. '/3

III

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B

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!

425

I

j

C'JlyqaHX,

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He

6y.n;eT

HMeT~CH

B'

BHAY

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COBepllIeHHO

onpf;).n;eJleHHoe

H3

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2(2),

2 H

T.

p;.

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cTaH,t:l;'aprHEleo603HaqemI:H.

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:Jil[t

t]

npHHHTO'

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npocTpaHcTBaMH

Ba-

Qt,

{3

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B

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aToro

IIaparpa.pa,

qTO,

KaK

H B

KOHeqHO-

MepHoM

npocTpaHcTBe,

MHomeCTBO

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(-t)

H3

JlHHeHHoro

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!

MHpoBaHHoro

npocTpaHcTBa

:Jil,

onpe.n;eJIeHHoe.

yc:'!oBHeM

p fhl

<:

e

(e>

0),

. (57.9)

\

I

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cif;epou

pap;Hyca'

e.

HanpHMep-,

B

rrpocTpaHCTBe

~

MHO-

t

metTBo

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,

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p

fh

- h(l)

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cil

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cljJepou

pa.n;Hyca

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ueHTpOM

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h(l).

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(T)

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sa

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eCJlH

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Tex

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6y-

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h ('t) .H·

HOpMhI

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h ('t)

~.

~

{hi

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p lh ('r)l, a

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-

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r

[h

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Ii

'I

~-,-,-_~_~~--,-,,-,-,--,-,,~.c..cc..:....c..:..--,-~'

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r /1, a 6 a 14

JIIIHElIHbIE

OIlEPAIJ;1I1I B UPOCTPAHCTBAX <l>YHRIJ;lIlI

§ 58.

HopMa

JIHHeHHOB

onepa~B

PaccMoTpHM

HeKOTopoe

.pYHKIJ;UOHaJII>HOe

JIUHeHHOe

HOpMupOBaH-

Hoe

npOCTpaHCTBO

f'iB[t""tfl]'

IIYCTI>

B

aTOM

npOCTpaHCTBe

onpeAeJIeHa

OnepaIJ;HH

q>

[h

('t')

1,

KOTopaH,

6YAY'llI

npOU3BeAeHa

HaA

.pyHKIJ;ue

H

h

('t')

H3

f'iB,

AaeT

B

pe3YJIbTaTe

Bem;eCTBeHHoe

'lHCJIO

1)

=

q>

[hI. (58.1)

fOBOpHT,

'ITO

q>

HBJIHeTCH

JIUHeHHOH

OnepaIJ;HeH

(UJIU

JIUHeHHbIM

epyHKIJ;UOHaJIOM

Ha

f'iB),

eCJIU

Bl>1nOJIHHIOTCH

CJIeAYIOm;ue

yCJIOBUH:

~

i)

(58.2)

q>

[MI

=

Aq>

[hI,

1)

','

I'·

KaKOBbl

61>1

HH

6bIJIH

'lHCJIO

AU

.pyHKIJ;UH

h

H3

f'iB;

~

,! 2)

q>

[h

ll

) + h(2l] =

q>

[h

ll

)I +

q>

[h(2)

l,

(58.3)

~

"

KaKOBbI

6bI

HH

61>1JIU

.pYHKIJ;lIH

h(l) U

h(2)

H3

fJa

(CM.

AJIH

cpaBHeHHH

~

,.

onpeAeJIeHUe

miHeHHoH

onepaIJ;uH

q>

lJ

n-MepHOM

npocTpaHcTBe,

§50,

I,

CTp.

397).

06Cy)l;UM

nOHHTHe

HOPMbI

JIHHeHHOH

onepaIJ;HH.

RaK

U B

KOHe'lHO-

~

MepHoM

CJIy'lae

(CM.

CTp.

398),

HopMa

p*

[q>1

onepaIJ;HU

<:p

onpe)l;e-

JIHeTCH

KaK

HaUMeHbmee

H3

'lHCeJI

v,

y)l;OBJIeTBOpHIOm;UX

HepaBeH-

!

CTBy

(58.4)

I

q>

[hI 1 < vp

[h

npH

h

H3

fJa.

OAHaKo

3)l;eCb

nOHBJIHeTCH

O)l;HO

HOBoe

cym;ecTBeHHoe

06cTOHTeJIbCTBo.

MOiKeT

CJIy'lHTbCH,

'ITO

HH

O)l;HOro

'IHCJIa

v,

KOTopoe

6bl

06eCne'lHBaJIO

HepaBeHcTBo

(58.4),

HeT.

B

COOTBeTCTBHH

C

aTHM

onpe)l;eJIHTb

HOpMy

p*

[q>

I

paBeHcTBoM,

aHaJIOrH'lHI>1M

paBeHcTBy

(51.3), ,o,JIli

JII060ll

JIHHeHHOH

onepaUHH

q>

Ha

f'iB[tl1.tsl

co

CBoHcTBaMH

(58.2), (58.3) B

06m;eM

CJIy'lae

YiKe

HeJIb311.

J];eJIo

B

TOM,

'ITO

BeJIH'IUHa

I

q>

[hI I

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He,o,OCTHraTb

KOHe'lHOrO

MaKcHMYMa'

Ha

c.pe-

pe

p [hI <

1.

1l0KaiKeM

9TO

Ha

npHMepe.

'

PaccMoTpHM

npocTpaHcTBo

'x[O,ll,

T.

e.

npocTpaHcTBo

fJa

.pyHKUHB

h

('t')

(0

<

't'

< 1), r,o,e

HopMa

p [hI

onpe,o,eJIeHa

paBeHcTBoM

(56.5).

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onepaIJ;HH

q>

[hI

conoCTaBJIHeT

KaiKAoH

.pyHKIJ;HU

h

('t')

ee

....

.':',:i

p

>,:,.

§

58]

HOPMA

JIHHEf.!HOf.!

OITEPAUHH

427

ilHa'IeHHe

B

TO'IKe

t =

0,

T.

e.

cp

[hI = h (0). (58.5)

Onepan;HH

cp

[hI (58.5)

YAOBJIeTBOpHeT

yCJIOBHJIM

(58.2) U (58.3).

1l0cTpoHM

nOCJIeAOBaTeJIbHOCTb

.pyHKUHH

h(i)('t')

(i

= 1, 2,

...

)

CJIe-

AYIOm;ero

BH)l;a:

1

i

npH

O<'1'<-i-'

h(i)('1')

=

(58.6)

1

-.

<'1'<1.

I

°

npH

I

HeTpYAHO

nOAC'lHTaTb,

'ITO

I

P [h(i)] =

~

1h(i)

('1')

I

d'1'

= 1.

(58.7)

°

CJIe,o,OBaTeJIbHO,

Bce

.pYHKIJ;HHh(i)

('t')

(58.6)

COAepiKaTCH

B

e)l;lIHH'lHOH

c.pepe.

B

COOTBeTCTBHH

C (58.5) H (58.6)

HMeeM:

q>

[h(i)] = h(i) (0) = i.

3TO

03Ha'laeT,

'ITO

lim

q>

[h(i)] =

00,

;""'00

T.

e.

onepaUHH

q>

(58.5)

He

HMeeT

Ha

e)l;HHH'lHOH

c.pepe

KOHe'IHOrO

MaKcHMyMa.

3Ha'IeHHH

paCCMOTpeHHOH

OnepaIJ;HII

Ha

9TOH

c.pepe

ae

OrpaHH'IeHhI

H,

CJIe,o,OBaTeJIbHO,

HeT

HH

OAHoro

'IIICJIa

v,

KOTopoe

YAO-

BJIeTBOpllJIO

6hl

HepaBeHcTBy

(58.4).

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06CTOJITeJIbCTBO

HBJIlieTCli

BnOJIHe

eCTeCTBeHHhIM,

TaK

KaK

onepaUHll

q>

(58.5)

cono-

CTaBJIHeT

!,aiK,o,oM

.pyHKUHH

h

('t')

113

'x[O,I]

ee

3Ha~leHHe

B O)l;HOH

TO'IKe

't'

=

O.

A

3HatIeHlle

.pyHKUHH

B

OAHOH

oT)l;eJIbHOH

TO'IKe,

KaK

6hIJIO

OT-

Me'IeHO

BhIllle

(CM.

§

57),

HBJIlieTCli

l-lecyu?eCmeeWibLM,

eCJIH

<PYHKUHIO

9TY

paCCMaTpllBaTh

KaK

ClJIeMeHT

npOCTpaHCTBa

'x[O,t].

YKa3aHHoe

06CTOllTeJIbCTBO,

KOTopoe

B

KOHe'IHOMepHOM

CJIy'Iae

He

HMeeT

MeCTa,

Tpe6yeT

3)l;eCb

)l;onOJIHIITeJIbHorO

orpaHH'IeHHH

Ha

CBOllCTBa

onepaUHH

cpo

A

UMeHHO,

rOBOpJIT,

'ITO

JIHHeIIHaJI

onepaUHJI

cp

(58.1) - (58.3) OcpanUlie1-la,

eCJIU

ee

3HatIeHHll

1)

=

cp

[hI

Ha

e)l;H-

HHlJHOH

c<pepe

OrpaHH'IeHI>1,

T.

e.

cymecTByeT

'IHCJIO

v

TaKoe,

'ITO

I

CjJ

[hI I<v

npH

p [hl <

1.

(58.8)

B

KOHelJHOMepHOM

npOCTpaHCTBe

fJa

n

eC,H,K,aJL

JIHHeHHaH

OnepaIJ;HJI

<p

[h J

llBJUIeTCH

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B

)l;aJIbHeHmeM

H

AJIH

npOCTpaHCTB

fJa[t

t

6y,o,eM

paccMaTpHBaTb

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OrpaHH'IeHHbIe

OnepaIJ;HH

a,

13

J.

,<p,

OAHaKO

lJT06bI

He

neperpyiKaTb

TepMHHOJIOrHIO,

BCIOAY

B

'KHHre

B

nQHJITHe

«JIHHeHHaJI)}

OnepaIJ;H~

6YAeM

BKJIaAI>1BaTI>

CMhICJI;

·_···R7"""'.a£i!43'--·WA,,.~·-i-

i},""

"l·f"·,····;,:~··.fjt:····

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p,

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JIHHE:0:HbIE

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B

IIPOCTPAHCTBAX

(])YHKnHfl

[rJI.

14

428

u

.~

.

oapaHutteHHafl,

JIliHeHHaJl

orrepa.u;HH,

QrrycKHH

CJIOBO

«OrpaHH'IeHHaH».

OAHaKO

Aame

eCJIH

orrepa.u;HH

ep

[hI

or{laHH'IeHa,

onpeACJIHTb

HOpMy

p*

[ep]

paBeHcTBoM

(51.3)

Bce-TaKH

MomBO

He

BcerAa,

TaK

KaK

MomeT

He

cym;ecTBoBaTb

TaKoro

3JIeMeHTa

h HS C«pepbl P [hI <1,

Ha

KOTO-

pOM

oocmuaaemcfl,

HaHOOJIbmee

SHa'IeHHe

I

ep

[h]l,

B

'IeM

Hac

yoeiR~

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CJIeAYIOm;HR

npHMep.

PaCCMOTpHM

«PYBK.u;HIO u ('r)

(58.1)

TaK:

'1']

CHOBa

rrpOCTpaHCTBO

«pyHK.u;HR

2[0,1]

H

=

-r

(0 <

-r

< 1).

OrrpeAeJIHM

JIHHeRHYIO

1 1

=

ep

[hI

=~h(t)

u

(t)d-r

=

~h(t)

-rdt.

° °

BbloepeM

orrepa.u;Hlo

(58.9)

O'IeBHAHO,

'ITO

AJIH

JIIOOOR «pyHK.u;HH h

(-r)

HS

2[0,1]'

JIemam;eR

B

c«pepe

p [hI <1,

cnpaBeAJIliBO

HepaBeHCTBO

PaCCMOTpHM

~

it

'\

rAe

B , 8

"",

I:i

1

2

nOJIOfKHTeJIbHbIX

:~!

~

1']

=

,--

ep

[hI =

1

~

h

(r)

t

d-r

<1.

(58.10)

°

rrOCJIeAOBaTeJIbHOCTb

«pyHK.u;HR

( 0,

0<

t'

<1 -

Bil

h(i)

(t)

= 1

1

(58.11)

-

1-8·<t<1

e. '

'L

--....;::

-....;::'

•

Bi,'"

- CXOAHm;aHCJI K

aymo

nOCJIeAOBaTeJIbHOCTb

'IHCeJI.

O'IeBHAHO,

'ITO

1

p[h(i)] =

~lh(i)(t)ldt=1,

°

T.

e.

Bce

«pyHK.u;HH (58.11)

COAepll{aTCH

B

eAHHH'IHOR

c«pepe.

HaRAeM

Tenepb

SHa'IeHHH

1']i

JIHHeRHOR

orrepa.u;HH

(58.9)

Ha

«pyHK.u;HHX

h(i)(-r)

(58.11).

MMeeM

1']i

= 1 -

8i/

2

.

Bce

CKasaHHoe

OSHa'IaeT,

'ITO

onepa.u;HH

(58.9) HBJIHeTCJI

orpaHH-

'IeHHOR

H

sup

ep

= 1

AOCTHraeTCH

Ha

«pyHK.u;HHX

h(i)

(t)

(58.11)

npH

8i

-+

0, HO

COrJIaCHO

(58.10)

He

cym;ecTByeT

«pyHK.u;HH

hO

(-r)

HS (58.11),

rAe

ObIJIO

Obi

ep

[hal

=

sup

ep

[h(i)] = 1.

B

CHJIy

HSJIOmeHHoro

HOpMy

OrpaHH'IeHHOR

JIHHeRHOR

orrepa.u;HH

<p

[hI

onpeAeJIHIOT

B

«PYHK.u;HoHaJIbHbIX

rrpocTpaHcTBax

5i3

paBeHcTBoM

p*

[ep]

=

sup

(\

ep

[h]1

rrpH

p [hI < 1). (58.12)

3Aecb

SHaK

sup

OSHa'IaeT,

'ITO

p*

[ep]

eCTb

BepXHHH

rpaUb

MHOiReCTBa

'IHOeJI

1']

= ep[hl,

nOJIy'IaIOm;HXCJI,

KorAa

h

rrpooeraeT

eAHHH1IHYIO

c«pepy.

B

'IaCTHOCTH,

AJIH

orrepal'l;HH

(58.9)

HMeeM

p*

[ep]

= 1.

o

§

59]

OIIEPAIJ;HH

B

CTAH.D:APTHbIX

HPOCTPAHCTBAX

429

B

SaKJIIO'IeHHe

OTMeTHM

CJIeAYIOm;ee

BaiRHOe

OOCTOHTeJIbCTBO.

RaK

H B

KOHe'IHOMepHOM

CJIy'Iae,

HOpMy

onepa.u;HH

ep

[hI

'IaCTO

YAOO-

HO'BbI'IHCJIHTb

HS

paBeHCTBa

1

p'[ep]

=-0

•

(58.13)

p

rAe

pO

ecTb

paCCTOHHHe

OT

TO'IKH

h = 0 AO

nOBepXHOCTH

ep

[hI =

1,

T.

e.

pO

=

inf

p [hI

rrpH

ep

[hI = 1.

t(58.14)

reOMeTpH'IeCKaH

HHTepnpeTa.u;HH

:noro

COOTHomeHHH

B

KOHeqHO-

MepHOM

cJIy4ae

npHBeAeHa

B § 52.

CTporHR

BbIBOA

COQTHOmeHHR

(58.13) H (58.14)

qHTaTeJIb

MOiReT

HaRTH

B

KHHre

[24*].

§

59.

JIHHeHHble

0!lepan;HH

B

cTaHAapTHbIX

npOCTpaHCTBax

- " (

'B

KOHeqHOMepHOM

BeKTopHOM

npOCTpaHCTBe

.;V'n

JIIOOaH

JIHHeR-

HaH

onepa.u;HH

ep

[hI

HMeeT

BHA

ep

[hI = u

1

h

1

+ u

2

h

2

+

...

+

unh

n

(CM.

TeopeMy

50.1). B OOm;eM

CJIyqae

«pyHK.u;HOHaJl);HOrO

npOCTpaH-

CTBa

5i3

[I

1

TaKOR

BCeOO'loeMJIIOm;eR

TeopeMbI,

KOToltaH

onpeAeJIHJIa

11,

~]

,

ObI

Ha

Bce

CJIyqaH

BHA

JIHHeRHOR

orrepan;HH

ep,

K

CO}KaJIeIDIIO,

HeT

•

0AHaKo

AJIH

MHorHX

KOHKpeTHbIX

«pYHKn;HOHaJIbHbIX

rrpOCTpaHCTB

5i3

[I

t

H,

B

qaCTHOCTH,AJIH

Tex

npOCTpaHCTB,

KOTopbIe

Mbl

oYAeM

11,

~]

HHme

paCCMaTpHBaTb,

MoryT

ObITb

yKasaHbI

«pOpMyJIbI,

onpeAeJIHIO-

owe

B

aTHX

npocTpaHcTBax

OOm;HR

BHA

JIHHeRHOR

onepan;HH.

IIpHBeAeM

SAeCb

oes

AOKaSaTeJIbCTBa

HeKOTopble

«paKTbI,

HSBeCT-

Hble

HS

«PYHK.u;HoHaJIbHorO

aHaJIHsa.

IIOAPooHoe

AOKaSaTeJIbCTBO

HX

MomHO

HaRTH

B

Kypcax

[14*,

20*1.

Teo

p

eM

a

59.1.

B

(/iyH"quoHa.l£bHQ.M,

npocmpaHcmee

2~n.t~]

(cmp. 417) o6UfuU euo

.l£UHeUHOU

onepal.fuu

ep

[hI

onpeOeAAemCfl,

c no-

MOUfblO

paeeHcmea

tlJ

ep

[hI =

~

h

(t)

u

(t)

dt,

(59.1)

t

l1

aoe

(/iyHK,qUfI,

U

(-r)

fl,8.I£MmCfl,

maK,JlCe

3.1£eMeHmOM

npocmpaHcmea

2f~~,t(3]'

HopMa

p*

[ep]

onepaquu

ep

(59,1) aaoaemcfl,

paeeHCmeOM

tlJ

.

'/

p.

[ep]

=

[~

u

2

(t)

dt]

"

(59.2)

t(1

16

H.

KpacOBcHiut

II

-~

..

----'---.!~.

"

..

>

..

,L...:..~,'

'>.'"

...

~i'.:~

...

··;··i·:···

..

'·:··l·,:':.··:.-;.,;··.····

~

..

'.

,'.'

.'lI'

"<-:.'"

,::""

...

, ...

:'.~

..

,

...

:;.

....

'::,;:,l.:.:.::;

..

:.-::;

.....

::.::::smr

"'J'

'.'

.,

·.'···:·:-·

~

o

""~,.-

..

'.'-

\(~:

..

;/

430

JIUHE11HbIE

OIIEPAIJ;MR

B

IIPOCTPAHCTBAX

YHKIJ;M11

[rJI.

14.

PaCCMOTpIlM,

HanpllMep,

npOCTpaHCTBO

4>YHKIJ;IlH

~f~~2"]

II

nOJIOmllMl

B (59.1) U

(T)

=

~

sin

T.

OnepaIJ;IlH

ep

aammIeTCH

Torp;a

TaR:

1t

2"

ep[k]

=+

~

k(T)sin-rd-r.

(59.3)

o

B peayJIbTaTe

onepaIJ;1l1l

(59.3)

MbI

nOJIy"laeM

K034>4>IlIJ;lleHT

<t>ypbe'

([14*),

CTp.

90)

4>YHRIJ;1l1l

k (T).

HeTpYP;Ho

BbI"lIlCJIIlTb

HOPMY

onepa-

IJ;1l1l

(59.3),

IlCXOP;H

HeIIocpep;CTBeHHO

llaonpep;eJIeHllH

HOpMbI

p*

[ep].

113

HepaBeHCTBa

EyHHRoBcRoro -

IIJBapIJ;a

(CM.

[14*],

CTp.

496)

tf!

'I

tf!

'/

I

~

k

(-r)

a

(-r)

d-r

I<

[~

k

2

(-r)

d-r

J

'.

[~

u

2

(1:)

dT]

,

(59.4)

ta. ta.

ta.

IlMeeM:

2"

'I

2"

'I

lep[k]I<[~

k

2

(T)d-rJ

'.

[~2

~sin2-rd1:]

'.

(59.5)

° °

Ma

aToro

HepaBeHcTBa

R

Ra

onpep;eJIeHIlH

BeJIIl"lIlHbI

p*

[ep]

(CM.

§

58,

I'

CTp.

428)

CJIe)J;yeT,

"lTO

J

2"

'/

P'[ep]<[~2

~

sin

2

-rd.]

'.

o

1 .

IIpll

aTOM

B (59.5)

Ha

3JIeMeHTeko

(T)

=

-Vii:

sin

T,

y)J;oBJIeTBOpHIOrn;eM

yCJIOBIlIO

p

[kO]

= 1,

oyp;eT

BhIlIOJIHHTbCH

paBeHCTBO.

II03ToMY

npH

h

(-r)

=

hO

(T)

IlMeeM:

2"

[k

O

[1.

] 1

('.

2 d 1 _ * [ ]

ep

] =

ep

-Vn

SIll

1:

= 1t

-Vit

~

SIll

•

-r

=

-Vi

-

pep.

°

PeaYJIbTaT,

eCTeCTBeHHO,

cOBn~)J;aeT

C

TeM,

"lTO

CJIe)J;yeT

113 (59.2).

IIoJIbaYHcb pe3YJIbTaTaMHBapllaIJ;1l0HHoro

IlC"lIlCJIeHllH

([4*],

CTp.

48; [10*],

MomHO

ep;HHcTBeHHbIH

3JIeMeHT

(c

HyJIeBOH

MephI)

ep;IlHIl"lHOH

nplI U

(-r)

=

~sin

T.

B

1t

1

nORa3aTh,

"lTO

kO

(T)

=

-Vii:

sin

-r

-

TO"lHOCTbIO

p;o

3Ha"leHIlH

Ha

MHomeCTBe

c4>epbI,

Ha

ROTOpOM

max

ep

=

ep

rho

I

OOrn;eM

CJIy"lae

onepaIJ;lllI (59.1),

Rorp;a

U

(T)

-

np01I3BOJIbHaH

HenpepbIBHaH

4>YHRIJ;IlH"

TaRme

cyrn;ecTByeT

ep;IlHCTBeHHltIH

aJIeMeHT

hO

(t),

JIemarn;IlH

B

ep;lIHIl"lHOH

c4>epe

II

y)J;O-

BJIeTBOpHIOrn;IlH

yCJIOBlIIO

max

ep

[hI =

ep

[hal

=

p*

[ep].

I

§

59]

OIIEPAnMM

B

CTAH)J;APTHbIX

IIPOCTPAHCTBAX

431

STOT

3JIeMeHT

BbI"llICJIHeTCH

no

4>opMYJIe

k

O ( ) _ u(·t'),

,_

u('t')

T - t

-.

I

f!

J'/,

p*[qJ]

[

Su

2

~

't') dt'

tex

.

ROTopaH

BbIBep;eHa

B

OCHOBHOM

TeRCTe

RHIlrlI npll penreHllIl

3ap;a"lH

00 ynpaBJIeHllIl B § 18.

IIepeHp;eM

R

npOCTpaHcTBy

~[ta..tl3}

Teo

p

eM

a 59.2. B

(/jYUlf,lfUOUa.l£bH,OM

npocmpaucmee

~[ta..tl3]

(CM.

CTp.

417)

o6UfUU

euf) .l£uueuuou Onepal{UU

ep

[hI

onpefJe.l£,f/,emc,f/,

paeeucmeoM (59.1),

ef)e

(/jYUlf,lfU,f/,

u.

(T)

,f/,6.I£.ftemc,f/,

a.aeMeumOM

npocm-

paucmea Jtt

(CM.

cTp.419).

HopMa

onepalfuu

ep

(59.1)

f)aemc,f/,

(/jOPMY-

.l£OU

1)

p*

[ep]

=

PaccMoTplIM

npllMep. IIycTb

4>YHRIJ;IlOHaJIbHOM

npOCTpaHCTBe

4>YHR~eH

U

(T),

MOP;YJIb

ROTOpOlI

IlMeeT

OP;IlH

1l30JIIl-

u

~

't'

t

8.·

Pnc. 59.1.

t

a

pOBaHHbIlI

MaRcllMyM

npll

£.

I

HeROTopOM

3Ha"leHlIIl

T = t I

(ta.

< t <

tfJ).

IIpllMeM

P;JIH

onpe)J;eJIeH-

HOCTlI,

"lTO

U (t) >

O.

B

3TOM

CJIy"lae

HeT

3JIeMeHTa

kO

(T),

ROTOpbIH

yP;OBJIeTBOpHJI

OhI

YC-

JIOBlIHM

p

[hal

<1,

ep

rho

(-r)]

=

max

I

ep

[hI I npll p [hI <

1,

0

RaR

3TO

OWIO

B

npep;bIp;yrn;eM

npllMepe

(CM.CTp.

430). JIerRo,

O)J;HaRO,

yRaaaTb

nOCJIep;OBa-

TeJIbHOCTb

4>YHRIJ;IlH

h(i)

(-r),

P;JIH

ROTOpbIX

IIplI i -+-

00

RMeeM:

.

Hruep

[h(i)

(.)]

=

sup

(I

ep

[hI I npll p [hI < 1) =

p*

[(pl.

i-->oo

3Ta

nOCJIep;OBaTeJIbHOCTb

TaROBa

(CM.

pRC.

59.1):

e.

0,

toc<,-r<t-+,

1

e. e.

k(i)(.)

=

~~,

t-+<-r<t+-y,

e.

0,

t++<-r<,t

ll

•

1)

eM.

CHOCKy

Ha

CTp.

419.

16-

sup IU

(T)I.

(59.6)

JIIlHeHHaH

onepaIJ;IlH

ep

(59.1) B

~[ta.,tI3]

nopomp;aeTCH

HeripepbIBHoH

r(i}('t'),urt'>

11

••

-.",'

.,

,',',,','

:

',-,-:"::.

".;.:.:::

d.·.· "':;.: ::::: , ,,:-::

':<-:.:

.,.'

'>':':.~~

p.

\2j;.

432

JIIDIE:fIHbIE

OIIEP.AIJ;IDI B IIPOCTPAHCTBAX YHHUH:fI

rrn.

14

B

CaMOM

JWJIi:)

~i

~

t+1!

ep[h(i)(t)J=

~'h(i)(t)u(t)dt=

~

+u(t)dr=u(t;),

tez

iE

i

t--

2

• [ e·

e· •

rp;e

t

i

-

TOqKa

H3

oTpe3Ka

t - -t '

t

+-t

J.Ho

lim

t

i

=t

rrpH

B.i~

0

:1

)I

0

11

0,

/

t;s-~

...

1

...

.1

IijTj-'"

/

llo(T)

t

lZ

\

)I

Pac.

59.2.

il

Torp;a

max

ep[h]

npH

yCJIOBHH p

[h]

< 1

6yp;eT

p;ocmraTbCH,

Ha-

'I

rrpHMep,

Ha

9JIeMeHT3

hO

(t)

=

11:

•

211:

(1'

- t )

~/.

SIn

a.

(59.7)

fl - ta.) tfl -

tao

'

cop;epmaID;eMCH B ep;HHHqHOH c41epe.

I1MeeM:

tfl-tez

~

211:

(1'

-

ta.>

dt

= v.

ep[hO]

=

~

tfl-tez

~

sin

tfl-

tao

tez

B OTJIHqHe

OT

cJIyqaa

rrpocTpaHcTBa

:t:;C2)

,

paccMoTpeHHoro,

BMIIIe,

9JIeMeHT

hO

('t) (59.7),

Ha

KO'l'OpOM p;OCTHraeTCH

max

ep

[h],

He

HB-

JIHeTCH ep;HHCTBeHHbIM (cYUfecme?Uuo).

HarrpIIMep,

B KaqeCTBe

TaKoro

9JIeMeHTa

hO

('t), rrOMnMO (59.7),

MomHO

ellW

paCCMOTpeTb 4lYHKIJ.;HlO

tfl-

tao

ta.-<t<--..,-

,

'hI)

('t) =

1_

"

;..

,

tfl --:

tao.

-<

t

-<

tfJ.

.

tfJ

-

tex.

'

·11

B 9TOM

CJIyqae

p*

{ep]

=

ep

Oi'oJ

= " (CM.

pHC.

59.2).

H

rr09ToMY

lim

ep

{h(i) ('t)] .

==u

(t)=maxlu

('t)/,

qTOHCJIe-

",

p;OBaJIO

rrOKa3aTb.

IIYCTb

Terrepb

U ('t) HBJIH-

eTCH

peJIeHHOH

4lYHKIJ.;HeH:

~

u

(t)

=

'(

\

hU(T)

/

tfl-t

",

/

ta.-<-r<

;..

ez,

'-'

=

1-

v,

U('()

tfl-

tao

-v,

--2--<t-<t

fl

•

F',

\(:,::,:

§

59]

OIIEPAUHH B CTAH,IJ;APTHbIX IIPOCTPAHCTBAX

433

PaCCMOTpHM

Terrepb

rrpOCTpaHCTBO <t6[ta..

tfl]'

Teo

p e M a 59.3. B

f/iYUK,l,fUOUa.I£bUOM

npocmpaucmee <t6[ta.. tfJ]

o6UfUU

eua

oapauulf,eH,UOU

Auueuuou

onepal,fuu

ep

[h]

ebLpaJICaemCll

uu-

meapaAoM

CmUAmbeCa

til

.

ep

[h] =

~

h ('t)

dU

(t),

(59.8)

ttt.

aae

U ('t) -

f/iYUK,4Ull

C

oapaUUlf,eH,Ubl.M,

U3.MeH,eHueM

(CM.

CTp.

45).

H

opMa amou onepa4uu

6bLlf,UC-

J1,R,(!mCll

no

f/iopMYAe

p.

[ep]

=

var

[U]t

(59.9)

a.

IIpocTpaHCTBO

4lYHKD;HH

U ('t) .

(ta.

-<

't

-<

tfl)

C

orpa-

't:

HHqeHHbIM H3MeHeHHeM

6yp;eM

o603Ha'IaTb

CHMBOJIOM 'Zr[tez.

tl>]'

Ie

(<DyHKUHH

U(l)

H

U(2),

OTJIH-

qalOID;HeCH

Ha

rrocToHHHyIO,

OTOmJIeCTBJIJIlOTCJI.)

PIIC.

59.3.

PaccMoTpHM

P;JIH

rrpHMepa

orrepaD;HlO

ep,

IIopomp;eHHylO

4lYHKD;HeH U ('t)

Bnp;a

(CM.

pnc.

59.3)

("11

tao

<'t<t

ll

IV2, t

1

-<

't <t

2

,

U

(t)

= \

..

t1l,-1

-<t-<t1l,'

V

n

+lI

t = t

n

=

tfl,

"n'

rp;e

Vi

>0 H

3HaKH

'Vi qepe,o;ylOTCH.

Torp;a

n

TJ

=

ep

{h] =

~

(Vi+!

-

Vi)

h

(ti)'

(59.10)

i=1

113 (59.10) CJIep;yeT, qTO

sup

11')

I

rrpH

p

[h]

=

max

lh

('t)

1-<16yp;eT

"

,o;OCTHraTbCH

Ha

JI1060H

Herrpep

blBHOH

4lyBKD;nH

h

Cr),

He

npeBlJma-

lOID;eH

rro

MO,o;yJIlO ep;HHHD;M H

rrpHHHMalOID;eH

B

TOqKaX

't=t

i

(i=1,

2,

...

,n)

9KCTpeMaJIbHMe

3HaqeHHH,

paBHbIe

h (t

i

)

=

(-1)i

(CM.

pHC.

59.3).

HopMa

p*

rep]

onepaD;HH

ep

B aTOM

CJIyqae

orrpep;eJIHeTCH

paBeH-

CTBOM

n

p.

[ep]

=

var

[U]:I> =

~

I

"i+1-Vi!'

ex.

i=1

••

7?

n '-t1T5l!G'6"wra-

'-n'rem-'a'

'iiYfW7

't'I>;;

ie,

{'n·"'·

-k

.........

Hi<"

.~

':f!;"

"_,'

........

","

it

" .

't·."."

"f'::

"--'-"-_

• •

.,,-'.

'c..'

·.·.c.:

.'.'

.:

~I

..:.

','

.,,:.

':-,'

",'

: :

:.,

..

' ,;;':," '

.'

"

"~H:-;~'~f:OB"f

c

434

llHHEnHME

OnEPA~H

B

npOCTPAHCTBAX

~YHH~Hn

[rJI.

14

CpaBImBaH

TeopeMM

59.1-59.3

C

tPaHTaMH,

npHBep;eHHhIMH

B

§§

50,

51

p;JUI

HOHe'lHOMepHhIX

npOCTpaHCTB,

y6emp;aeMCH,

lITO

aTH

TeopeMM,

eCTeCTBeHHO,

o606m;aIOT

yKaSaHHMe

peSYJII.TaTI.I.

IIpH

aTOM

JIHIIII.

onepaD;HH

CyMMHpoBaHHH

no

HHp;eKcy

i

(i

= 1, 2,..., n)

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no

't

(ta.

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a

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no

i

saMeItHeTCH

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:!

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rpaIm

no

't

(no

cYUfecmey).

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59.1-59.3

CJIe-

-;1

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onepaD;lIH

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(2(2),

..At

H

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me

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H B

KOHe'lHOMepHoM

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(CM. § 51),

npOCTpaHCTBO

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cOnpR,:HCelUf,b1Jr£

K

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[u]

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(HJIH

U)

HS

fJ3*

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conpHmeHHoH

K

HopMe

p [h],

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m;eH

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caMocOnpR,:HCeH-

Hb1Jr£,

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C HCXOP;HhIM

~

npocipaHcTBoM

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MomHo

nOCTaBHTI.

H

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sap;a'ly:

no

HeKoTopoMy

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pOBaHHoMy

tPYHKD;HOHaJII.HoMY

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('t)}

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Koe

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('t)},

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('t)}

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::hy

B

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sap;a'ly

MbI

sp;ecI.

He

paCCMaTpHBaeM.

OTMeTHM

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C

npOCTMMH

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aTOH

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MbI

BCTpe-

II!

I

THMCH

npH

peIIIeImH

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sap;a'l

06

ynpaBJIeImH,

rp;e

H

npH-

Bep;eHhI

HYmHhle

p;aHHMe.

lIs

npHBep;eHHMX

BMIIIe

TeopeM

CJIep;yeT,

'ITO:

1)

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2(2) -

caMocoIIpHmeHHoe

npocTpaHcTBo;

2)

npOCTpaHCTBO

tPYHKD;HH U (t) HS

..At

eCTI.

conpHmeHHoe

K

npo-

CTpaHcTBy

tPYHKD;HH h

(t)

HS

2;

3)

npOCTpaHCTBO

'Zl'tPyHKIIHH

U

(t)

C

OrpaHH'leHHMM

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HBJIHeTCH

conpHmeHHMM

K

npOCTpaHcTBy

~

HenpepMBHMX

tPYHK-

D;HH h (t).

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em;e

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onepaD;HH

B

rHJII.6epTOBoM

tPYHKD;HOHaJII.HOM

npOCTpaHCTBe.

,IJ;oKaSaTeJII.CTBO

ee

TaKme

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B

CTaHp;apTHhIxKypcax

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[14*]).

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p

eM

a 59.4. B

rjiYH1f,LfUOHa.t£bHOM

zU.fI,b6epmOBoM

npocmpaH-

cmBe

;;t[ta.. t

p

]

o6UfuU

BUO

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ep

[h]

onpeoe.fI,MmcR,

c

nOMoUfblO

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ep

[h] = (h (t), U (t)), (59.11)

zoe

U (t)

R,B.fI,R,emCR,

ma1f,:HCe

9.f1,e.MeHmO.M

ua

;;t[ta..

t(3]'

a

CU,M,BO.fI,

Buoa (h

y

(t),

h

s

(t)) OaHa'laem

C1f,a.fl,R,pHOe

npOU3BeOeHUe,

onpeoe.fl,R,lO~e

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()

l'

60]

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B

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435

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p*

[ep]

onepaLfuu

<p

[h]

aa-

oaemcR,

paBeHCm6oJrt,

p*

[<p]

= (u, u)'f,.

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B

Ka'leCTBe

npOCTpaHCTBa

;;t[ta..

t(3]

B

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59.4

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t(3]'

TO

nOJIy'lHM

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59.1.

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peMa

59.4

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0606m;eImeM

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59.1

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npOCTpaH-

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TOBO

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caMoconpHmeHHhIM.

)l;JIH

rHJII.6epTOBa

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;;t,

onHcaHHoro

B

npHMepe

6°

§ 56,

nOJIy'laeM

TenepI.,

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06m;HH

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onepaD;HH

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[h]

B aTOM

.

npOCTpaHCTBe

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t(3

tfJ

t(3

<p[h]

=

~

~J(('t,tJ)h,(

..

)u(tJ)d't'dtJ+

~h'(tJ)u(tJ)dtJ.

ta.

ta. ta.

3p;ecI.

tPYHKD;HH U

(tJ),

HBJIHIOm;aHCH

TaKme

aJIeMeHTOM

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;;tUa.,t(3]

, H

onpep;eJIHeT

onepaD;HIO

<p

[hl.

IIpH

aTOM

oIIepaD;HH

ep

[hJ

IHMeeT

HOpMy

t(3

til

t(3

p*[<p]

=[~

~J(('t',tJ)u('t')u(tJ)d

..

d'&+

~u2(tJ)dtJr.

ta.

ta. ta.

§ 60.

JIHHeiiHI.Ie

OIIepaIIHII

B

npocTpaHcTBax

BeKTop-epyHKIIiI'ii

PesyJII.TaTbI,

nplIBep;eHHhle

B § 59,

nepeHOCHTCH

H

Ha

60JIee

06m;HH

oCJIyqaH tPYHKIJ;HOHaJII.HMX

npOCTpaHCTB,

a

HMeHHO,

Ha

CJIy'laH

npo-

oCTpaHCTB

fJ3,

aJIeMeHTaMH

KOTOpMX

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iQHH h ('t) = {hi ('t)}

(ta.

-<

't

~

t~).

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nocTpoeHHH

pasJIH'lHMx

KOHKpeTHMX

HOpM

p [h] MomHO BOCnOJII.SOBaTI.CH CJIep;yIOm;HMH

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pameImHMH.

IIYCTI.

y

Hac

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r-MepHMX

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HOM

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't

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I, n

lIS

§ 49.

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JII060M

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't

MHomeCTBO

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h ('t)

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npocTpaHcTBoM.

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MI.I

MO-

:meM

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r

[h

('t)]

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lIS

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(CM.

§ 49).

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r

[h

('t)]

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tPYHKIJ;IIeii

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HII

't,

saBIICHm;eH

01'

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h ('t'). /

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tPYHKIJ;IIei

r [h

(or)]

p;eHCTBHH,

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(57.3), (57.5), (57.6), (57.8), (6.8),

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p [h] B

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p .

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CTBa

fB,

paccMoTpeHHLIe B § 57.

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et

< T < til)

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l.

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MLI

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,IJ;aJIee,

p [h) =

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437

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r

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et

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(1'1£+1)

- U

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=

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CTp.

400, § 59, (59.2»,

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(60.14).

t=l

06IIJ;HH

BHA

JIHHeRHOR

orrepan;HH

<p

[h] B

TaROM

rrpocTpaHcTBe 6YAeT aaAa-

BaTMJI BHpameHHeM

(60.3)

(CM.

CJIY'IaH

II

Ha CTp. 436)

tfj

r

<P

[h]

=

~

h'

(or)

dU

(or)

=

~

\

(or)

dU

i

(or).

ta.

i=l

,!l;aJIee, Ha (60.4) H (60.7) CJIeAyeT, 'ITO

tfj

r II,

~

r

~

r

~

p.

[U]=

~

[~

dU;]

.

(60.15)

<U

(2]

hr(or»)dor]

[~(~

u~(or»)dT]'

(60.10)

2.

t",

i=l

t",

i=l

t",

i=l

I

IIpH

He

OCTaHaBJIHBaJICh

Ha

APyrHx

BoaMOlHHbIX

npHMepax,

OTMeTHM,

'ITO B

§§

12,

I

14:

npHBOAJITCJI

Ta6JIHn;hI 12.1 H

14.1,

rAe

yRaaaH

Ha60p

HOpM

p [h] H

p*

[u]

HJIH

p*

[dU],

ROTopbIe

OKaabIBaIOTCJI

nOJIeaHbIMH

npH

HaY'IeHHH aaAa'I

06

ynpaBJIe-

h.

(or)

_ u

i

(T)

(60.11)

1 - til r

ffim

H Ha6JIIOAeHHH.

[L

(~l

ur

(or»)

dorf'

§ 61.

3aJ(a'Ia

00

OTJ(eJICHIUf

BblIIYRJIbIX

MHO'I{eCTB

I

B (60.10) 6YAeM HMeTh paBeHCTBO, 'ITO H AORaabIBaeT crrpaBeAJIHBOCTh

COOTHome-

!

B §

53

MhI

paCCMaTpUBaJIH

3aJ(a'lY

00

OTp;eJIeHHH

BhIIIYKJIhIX

MHO-

HHJI

(60.9).

IIPH

BTOM

TaR

me,

RaR H B

rrpOCTpaHCTBe

2(Z)

(CM.

§ 59). BeRTOp-

meCTB

B

KOHe'IHOMepHhIX

BeKTopHhIX

npOCTpaHCTBax.

B

HaCTOHIIJ;eM

cPYHKn;lIJI

h

(or)

(60.11)

JIBJIJIeTCJI

eAHHCTBeHHbIM

:,meMeHTOM

Ha

eAlIHH'IHOH clflephI,

Ha

ROTOpOM

AOCTHraeTCJI

max

<p

[h]

.

naparpawe

MhI

OOCYP;HM

aTy

3Up;atIY

B

CJIYtIae

oeCROHetIHOMepHoro

IIplHlep

60.2.

IIYCTh B rrpocTpaHcTBe BeRTopoB rrpH Ralf\AOM T HopMa y [h

(or)]

epYHKD;HOHaJIhHOrO

npOCTpaHCTBa

.A!'.

orrpeAeJIeHa paBeHcTBoM (60.5).

IIpon;eAYPY

X

[y]

BbI6epeM B

COOTBeTCTBHH

C

IfTaR,

paCCMOTpHM

HeROTopoe

JIHHeHHOe

WYHRD;HOHaJIbHOe

npo-

COOTHomeHHeM

(56.5),

orrpeAeJIJIIOIIJ;HM

HOPMY

B npOCTpaHCTBe

2,

T.

e.

CTpaHCTBO

.A!'[ta..

t

{3h

8JIeMeHTaMH

KOToporo

llBJIJIIOTCH

WYHKD;HH

HJIH

~ ~

r

~

BeRTop-WYHRIJ;HH

h

(T),

3aJ(aHHhle

Ha

OTpe3Re

[ta.,

t{3].

BBep;eM

onpe-

p [h] = X

[r]

=

~

r

[h

(or)]

dor

=

~

(~hr

(or»)

'dor.

(60.12)

,n;eJIeHHe

BhInYRJIOrO

MHOlReCTBa

(CM.

§ 53,

CTp.

404).

t", t",

i=l

IIycTb

·h(l)

(T)

H

h(Z)

(T)

-

8JIeMeHThI

H3

.A!'.

PaccMoTpHMBce

epYHRIJ;HH h

(T)

BHJ(a

06IIJ;HR BlIA

JIHHeHHOH

onepan;HH

<p

[h]

B

TaROM

rrpocTpaHcTBe 6YAeT orrpeAe-

JIJIThCJI

BbIpameHHeM (60.1)

(CM.

CJIY'IaR I Ha CTp. 436).

IIa

(60.7), (59.13) H (60.2)

heAl

(T)

=

"A

h(l)(T) +

(1

-

"A)h(Z)(T)

(0

<"

"A

<"

1). (61.1)

HMeeM:

r

tl

RaR

H B

CJIYtIae

ROHetIHOMepHoro

npOCTpaHCTBa

.A!'

n,

MHomeCTBO

WYHR-

X*

[r*]=sup

[~

Ur(T)]

•

=p*

[<pl.

(60.13)

IJ;HH (61.1)

MomHO

TpaRTOBaTb

RaR

«OTpe30R»,

coep;HHHIOIIJ;HH

B

npo-

r

t=l

CTpaHCTBe

.A!'

TO'IRH

h(l)(T) H

h(Z)(T).

HanpHMep,

nYCTb

h(l)

(T),

h(2)(T)

(ta.

<"

T

<"

t{3)

-

8JIeMeHThI

IIpOBepRY

COOTHOmeHHJI

(60.13)

AJIJI

HOpMbI

p*

[<p]

JUlHeRHoH orrepan;HH

lp

npOCTpaHCTBa

YfJ[ta..

t

{3]

HenpephIBHhIX

WYHRIJ;HH.

fpaWHRH

aTHX

1IIhI

aAeCh

nOAP06HO

npOBOp;HTh

He 6YAeM.

OTMeTHM

TOJIhRO,

'ITO cnpaBeAJIHBOCTb

(60.13)

MOlf\HO npOBepHTh TaR

me,

RaR H B rrpHMepe 60.1.

4YHKIJ;HH

1I300pameHhI

Ha

pHC.

61.1. .

IIpHMep

60.3.

IIYCTh

lflYHKn;HH

hi

(T) - HenpepbIBHbI H HopMa y [h

(or)]

Torp;a

rpaWHRlI

WYHRIJ;HH

h ('t),

COCTaBJIfnOm;He

OTpe30R

(61.1),

lleRTopa

h

(or)

= {hi

(or)}

orrpeAeJIeHa paBeHcTBoM (60.5). IIpon;eAYPY X

[V]

BH'

I

I'

aanOJIHHIOT

oOJIacTb,

3anITpHXOBaHHYIO

Ha

pHC.

61.1.

Красовский Н.Н. Теория управления движением

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